P. Van Emde Boas, Another NP-complete partition problem and the complexity of computing short vectors in a lattice, Tech. Report 81-04, Dept. of Mathematics, Univ. of Amsterdam, 1980.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....The question of finding the shortest non zero vector in a lattice with repsect to the L1 was proved to be NP hard by Van Emde Boas. However the corresponding problem for the L 2 norm (or any other L p norms for 1 p 1) remained unsolved. Van Emde Boas conjectured almost twenty years ago (cf. [vEB]) that the L 2 shortest vector problem for lattices in Z n is NP hard and the corresponding decision problem is NP complete. The ff approximate version of the problem is the following: find a non zero vector v in the lattice L so that its length is at most ffkv 0 k where v 0 is a shortest ....

P. Van Emde Boas, Another NP-complete partition problem and the complexity of computing short vectors in a lattice, Tech. Report 81-04, Dept. of Mathematics, Univ. of Amsterdam, 1980.

....easily verified to be sympletic. Then A Q = n # i =1 Q(# i )Q(# # i ) 0. The contrary is trivial since by the classification theorem Q # = n #H 0 . 5 Computational methods for quadratic forms References: Lag59, Jac57a, Jac57b, Gun81, Fro94, Fro06, LT85, Bri56, Leb56] References: [Hay68, Cot74, BP71, BKP76, BK77, BG, LLL82, Sch84, Hel85a, Hel85b, Bab85, Kan83a, Kan83b, Kan86, Lag80, Sch86a, Sch86b, vEB81, BK84, Die75, BK79, Poh81, AG85, Kan, FP85, Kal83] 6 Exercises 22 6 Exercises 1 Show that the group of self isomorphisms of the bilinear form given by the ndimensional identity matrix is isomorphic to the group of n n orthogonal matrix over Z. 2 Prove that the following statements are equivalent: 1. #(#,#) 0 for all # # M , then # ....

P. van Emde Boas. Another NP-complete partition problem and the complexity of computing short vectors in a lattice. Rept. 81-04, Dept. of Mathematics, Univ. of Amsterdam, 1981. 21

....closest vector problem to the shortest vector problem. The closest vector problem is de ned as follows. Given n linearly independent vectors b 1 ; b n 2 Q n , and a further vector q 2 Q n , nd a vector x in the lattice generated by b 1 ; b n with jjq xjj minimal. Van Emde Boas [96] showed that nding the shortest vector with respect to the maximum norm in a given lattice is NP hard, and that the closest vector problem is NPhard for any norm. Just as the rst basis vector is an approximation of the shortest vector of the lattice (14) the other basis vectors are ....

P. van Emde Boas (1981), Another NP-complete partition problem and the complexity of computing short vectors in a lattice, Report 81-04, Mathematical Institute, University of Amsterdam, Amsterdam.

....closest vector problem to the shortest vector problem. The closest vector problem is de ned as follows. Given n linearly independent vectors a 1 ; a n 2 Q n , and a further vector b 2 Q n , nd a vector x in the lattice generated by a 1 ; a n with jjb xjj minimal. Van Emde Boas [15] showed that nding the shortest vector with respect to the maximum norm in a given lattice is NP hard, and that the closest vector problem is NPhard for any norm. Just as the rst basis vector is an approximation of the shortest vector of the lattice (14) the other basis vectors are ....

P. van Emde Boas (1981). Another NP-complete partition problem and the complexity of computing short vectors in a lattice. Report 81-04, Mathematical Institute, University of Amsterdam, Amsterdam.

....yet not a proof, that many of the well known algorithmic problems for lattices are computationally hard for P. Regarding NP hardness, Lagarias [16] showed that the shortest vector problem is NPhard for the l 1 norm, but it is not known whether it is NP hard under any other l p norm. Van Emde Boas [24] showed that finding the nearest vector is NP hard under all l p norms, p 1. From [3] it is known that finding an approximate solution to within any constant factor for the nearest vector problem for any l p norm, and, for the shortest vector problem in the l 1 norm, are both NP hard. There are ....

P. van Emde Boas. Another NP-complete partition problem and the complexity of computing short vectors in lattices. Technical Report 81-04, Mathematics Department, University of Amsterdam, 1981.

....that some versions of this problem are NP hard. In contrast, neither is known to hold for factoring, and for discrete log the usual random self reducibility is only valid for a fixed modulus p. Regarding NP hardness, Lagarias [Lag82] showed that SVP is NP hard for the l 1 norm. Van Emde Boas [vEB81] showed that finding the nearest lattice vector is NP hard under all l p norms, p 1. Arora et al. [ABSS93] showed that finding an approximate solution to within any constant factor for the nearest vector problem for any l p norm, is NP hard. There are no known polynomial time algorithms to find ....

P. van Emde Boas. Another NP-complete partition problem and the complexity of computing short vectors in lattices. Technical Report 81-04, Mathematics Department, University of Amsterdam, 1981.

....the problem that for any norm k k, min k b Gamma Ax k (3) x 2 R n integral; where b 2 R m , and A is an m Thetan matrix with integer elements. This problem, called the closest vector problem in integer programming, has been proven to be NP complete even for simple norms such as l 2 and l 1 [11, 24, 25]. Another example is related to the solution of a class of more general problems: mixed integer nonlinear programming problems. A mixed integer nonlinear program min g(x; y) 4) y 2 R m x 2 R n integral can be formulated, under appropriate assumptions, as a nonlinear integer program min ....

P. van Emde-Boas [1981]. Another NP-Complete Partition Problem and the Complexity of Computing Short Vectors in a Lattice. Report 81-04, Mathematical Institute, Univ. of Amsterdam, Amsterdam.

....The issue of nontrivial rounding strategies in representing geometric objects has not had much attention in the literature. The simple decomposition of Section 2 is surely not new, but we could not find it in the literature. Section 3 relates the literature on closest lattice point problems [1, 10, 13, 14, 19] to the rounding application. Determining the optimal free knot spline has received considerable attention [3, 7, 16] and we take advantage of that work. Some details of our spline optimization, B spline differentiation, and Hausdorff distance computation may be of interest even to people working ....

....Rx 0 . The set of all such linear combinations is a subset of real n space R n closed under addition. Such subsets are called integer lattices and the problem of minimizing kR(x Gamma x 0 )k 2 is called the closest lattice point problem. Van Emde Boas has shown that this problem is NP complete [19], but good approximate solutions can often be found using the Lov asz lattice basis reduction algorithm[10] The idea is to use the Lov asz algorithm to find an alternative representation of the lattice that makes it easier to find a lattice point close to Rx 0 . Originally, the lattice is ....

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P. van Emde Boas, Another NP-complete partition problem and the complexity of computing short vectors in a lattice, Report 81-04, Math. Institute, Univ. of Amsterdam, 1981.

....; x n )k 1 = max 1jn jx j j; then we find that the vector e 0 has norm 1 2. Therefore, we can solve all subset sum problems of any density if we have a lattice oracle for the sup norm, as was pointed out by Michael Kaib. The general sup norm shortest vector problem is known to be NP complete [7]; the complexity of the square norm shortest vector problem is an open problem. That a sup norm lattice oracle yields a better density bound than a square norm lattice oracle suggests that the shortest vector problem for the sup norm might be harder than for the square norm. The discussion above ....

P. van Emde Boas, Another NP-complete partition problem and the complexity of computing short vectors in a lattice, Rept. 81-04, Dept. of Mathematics, Univ. of Amsterdam, 1981.

....of R as shown in Figure 34. With the block structure shown in the figure, the problem reduces to finding an integer vector VP so as to minimize kRPP VP RP1V 1 k (23) RPP RP1 0 VP V 1 Figure 34: Block structure for the reduced problem Van Emde Boas has shown that this problem is NP complete [17], but good approximate solutions can often be found using the Lov asz lattice basis reduction algorithm. 13] See also Babai [3] for more details and an analysis of the approximation algorithm) When applied to RPP , the algorithm finds a transformed matrix RPP T such that the matrices T and T ....

P. van Emde Boas. Another NP-complete partition problem and the complexity of computing short vectors in a lattice. Report 81-04, Math. Institute, Univ. of Amsterdam, 1981.

....NP hard, where fl is an arbitrary small positive constant and n is the size of the MinDES1 instance. 5. 3 Aggregation Part II The following Lemma, originally proved by Anthonisse [1] is a slight variation of the former Lemma and crucial for our reduction; for the simple proof see also [18, 16]. Lemma 16. Let A be an integral 2 Theta n matrix and b 2 Z. Then B n x 2 Z n fi fi fi Ax = h b 0 io = B ae x 2 Z n fi fi fi fi n X j=1 (a 1;j ka 2;j )x j = b oe where B denotes the n dimensional ball of 1 radius centered at the origin and k P n j=1 ja 1;j ....

P. van Emde Boas. Another NP-complete partition problem and the complexity of computing short vectors in a lattice. Technical Report 81-04, Math. Inst., University of Amsterdam, 1981. This article was processed using the L a T E X macro package with LLNCS style

....; x n )k 1 = max 1jn jx j j; then we find that the vector e 0 has norm 1 2. Therefore, we can solve all subset sum problems of any density if we have a lattice oracle for the sup norm, as was pointed out by Michael Kaib. The general sup norm shortest vector problem is known to be NP complete [7]; the complexity of the square norm shortest vector problem is an open problem. That a sup norm lattice oracle yields a better density bound than a square norm lattice oracle suggests that the shortest vector problem for the sup norm might be harder than for the square norm. The discussion above ....

P. van Emde Boas, Another NP-complete partition problem and the complexity of computing short vectors in a lattice, Rept. 81-04, Dept. of Mathematics, Univ. of Amsterdam, 1981.

....be applied to compute minimal polynomials of an algebraic number, simultaneous) diophantine approximations and integer dependencies among real vectors (see [12, 13, 10] Obviously, for a non zero vector x 2 Q n there are n Gamma 1 linearly independent integer relations. However, van Emde Boas [16] has shown that the decision variant of SIR1 is NP complete. For arbitrary real non zero x 2 R n it cannot even be decided in a very general model of computation whether there exists an integer relation at all (see Babai, Just and Meyer auf der Heide [7] On the other hand, Hastad, Just, ....

.... SV p for integral lattices in the same p norm, i.e. the problem of finding for an integral basis b 1 ; bn of an additive subgroup of the Z n , the p shortest non zero linear integral combination of b 1 ; bn ; its decision variant is known to be NP complete for p = 1 (see [16]) On the other hand, Arora, Babai, Stern and Sweedyk [4] have shown that under the widely believed assumption NP 6 QP there exists no polynomial time algorithm approximating the Shortest Vector problem in the 1 norm within a factor of 2 log 0:5 Gammafl n , where fl is an arbitrarily small ....

P. van Emde Boas. Another NP-complete partition problem and the complexity of computing short vectors in a lattice. Technical Report 81-04, Math. Inst., University of Amsterdam, 1981.

.... Integer Relation in 1 norm (SIR1 ) INSTANCE: A rational vector a 2 Q d SOLUTION: A nonzero vector x 2 Z d such that ha; xi = 0 MEASURE: The 1 norm kxk1 : max 1in jx i j of the vector x The Shortest Integer Relation problem in 1 norm was proven to be NP complete by van Emde Boas [19]. Very recently, Rossner and Seifert [16] showed the following Theorem, stating that it is even almost NP hard to approximate SIR1 in polynomial time within a factor 2 log 0:5 Gammafl n , where fl is an arbitrary small positive constant and n the size of the SIR1 instance I . Theorem 4. ....

P. van Emde Boas. Another NP-complete partition problem and the complexity of computing short vectors in a lattice. Technical Report 81-04, Math. Inst., University of Amsterdam, 1981.

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